The cartridge belt was rarely used for cartridges; usually it held cigarettes or C-Ration chocolate bars. .30-'06 Ammo was supplied in bright (later OD painted) "spam" cans packed two to a case. The ammo was in 8-round clips, in bandoliers. Troops generally just picked up a couple of bandoliers and slung them over their shoulders. It was common practice in actual fighting to have at least one clip in the pocket or under the helmet strap or even in the off hand ready for immediate use. I have heard of using the sling, but it seems to me not a very good way to carry a clip.
Ammunition issued in combat was almost always AP, the reason being that it gave better penetration in cover material (dirt, sand bags, etc.) and light vehicles than ball. Ball was usually issued only for stateside training.
When I first started shooting, a friend was an armorer in the Marine Reserve, and we (I mean he, of course) sometimes had to "test fire" rifles on the local range. Sometimes there would be WWII vets around, still young men at that time, and they loved to get their hands on an M1 again. Those who believe the silly story about the enemy hearing the ping of an empty clip and charging never watched a GI load an M1. Even aside from the others in his company, I guarantee no enemy was going to take more than a step or two before that empty rifle was ready to go again!